# Non-reduced locus of a locally Noetherian scheme is the closure of non-reduced associated points

Affine case (already proved by myself): The locus on $\operatorname{Spec} A$ of points where the stalks are nonreduced is the closure of those associated points of $\operatorname{Spec} A$ where the stalks are nonreduced.

I have trouble generalizing this to a locally Noetherian scheme $X$.

Let's first cover $X$ with open affines $U_i=\operatorname{Spec} A_i$ (this cover is not necessarily finite!). The associate points of $X$ are also associated in whatever $U_i$ containing them. The locus $L=\cup_iL_i$, where $L_i=U_i\cap L$. By the affine case, we know each $L_i$ is the closure those associated points of $U_i$ where the stalks are nonreduced. However, I don't know how to show that $L$ is the closure of associated points (it appears to be unions of closure instead of closure of union because of the lack of finiteness).

It is a pure topological fact that the closure of a set can be computed locally: if $$X$$ is a topological space, and $$U_i$$ is an open cover (not necessarily finite), and $$S$$ is any subset of $$X$$, then $$\bigcup_i((\overline{S\cap U_i})\cap U_i)=\overline{S}.$$ Obviously the right hand side contains the left hand side. On the other hand, if $$x\in\bar{S}$$, then assume $$x\in U_i$$, and $$x\in \overline{S\cap U_i}$$, so $$x$$ is contained in the left hand side.
Now apply this to the question above. Assume $$N$$ is the set of nonreduced associated points, and $$L$$ is the non-reduced locus in $$X$$, then by the affine cases, we know $$L\cap U_i=L_i$$ equals $$(\overline{N\cap U_i})\cap U_i$$. Then we can see $$\bigcup_i L_i=\bigcup_i (\overline{N\cap U_i})\cap U_i=\overline{N}$$ that is, the non-reduced locus is the closure of the set of nonreduced associated points.
Moreover, by the local finiteness of the set of associated points, we actually know any $$x\in X$$ is contained in the closure of the set of nonreduced associated points, if and only if it is contained in the closure of some non-reduced associated point $$p\in X$$.